Equator Conjecture for flag homology spheres

Let $\Delta$ be a flag homology sphere. An equator is an induced subcomplex that is a flag homology sphere of codimension $1$. Equator Conjecture. For every equator $E$ in $\Delta$, its $\gamma$-vector is coefficientwise at most that of $\Delta$:

$$\gamma(E)\leq\gamma(\Delta).$$

Every vertex link is an equator, and the supplied text states that this formal generalization is equivalent to the Link Conjecture. No resolution evidence is supplied.